Tunable Topological Phases in an Organic One-Dimensional Mott Chain: Odd-Haldane (S = 1/2) and Haldane (S = 1)

TL;DR

This work demonstrates that a single chemically realizable organic 1D chain can host two interacting symmetry-protected topological phases: a bond-centered odd-Haldane $S{=} frac{1}{2}$ chain and a site-centered Haldane $S{=}1$ chain built from Hund-coupled superatoms. By combining DFT+$U$ with a spin-only Hubbard mapping, ED, and DMRG, the authors extract robust exchange parameters in the deep Mott regime ($U/t \sim 100$–$130$) and identify characteristic SPT fingerprints, including a quantized many-body Zak phase, an almost fourfold entanglement spectrum, protected edge spins, and distinct triplon/Haldane spectra in $S^{+-}(q,\omega)$. The spin-$\tfrac{1}{2}$ chain shows a bond-centered odd-Haldane phase with edge $S{=}\tfrac{1}{2}$ states and a W-shaped triplon dispersion, while the Hund-coupled spin-$1$ chain realizes a Haldane phase with two edge $S{=}\tfrac{1}{2}$ spins and an M-shaped magnon spectrum with a gap around $\Delta E_H \approx 52$ meV. Finite-temperature molecular dynamics indicate only modest parameter renormalization up to 300 K, making this chemically programmable platform promising for edge-spin qubits and nanoscale quantum devices that exploit interacting SPT physics.

Abstract

Establishing symmetry-protected topological (SPT) phases with interactions in chemically realistic systems remains an open challenge. We show that a single, synthetically plausible organic one-dimensional chain, tunable via chemical modification of its radical sites, hosts two such phases: an odd-Haldane phase of a dimerized $S=\tfrac{1}{2}$ Heisenberg chain and a Haldane phase of an $S=1$ chain realized when Hund coupling locks two $S=\tfrac{1}{2}$ spins per monomer into $S=1$. Density-functional theory places the active manifold deep in the Mott regime ($U/t\!\approx\!126$), justifying a spin-only Heisenberg description; a compact $(t,U)\!\to\!J$ mapping then fixes exchange couplings. Exact diagonalization and DMRG reveal a consistent SPT fingerprint across both phases, including a quantized many-body Zak phase, even-degenerate entanglement spectrum, protected edge spins, and characteristic triplon/Haldane features in $S^{+-}(q,ω)$. Our results identify a chemically programmable molecular platform for interacting SPT physics in one dimension and suggest concrete spectroscopic routes to organic Haldane spin chains for nanoscale quantum devices.

Figures (4)

• Figure 1: Chain chemistry and terminations of the dimerized nanographene spin-$1/2$ chain. (a) Chemical route from N-doped CTP to a stable $S=1/2$ radical and its one-dimensional polymer, whose magnetic backbone is an alternating-exchange spin-$1/2$ chain with couplings $J_1$ (blue) and $J_2$ (red). (b) Simulated spin-polarized STM constant-current maps for the isolated radical (top: $\alpha$-spin, bottom: $\beta$-spin), showing complementary contrast at the same molecular site (red arrows), consistent with a single $S=1/2$ ground state. (c) DFT+$U$ local magnetization for a polymer segment, revealing bulk antiferromagnetic order with alternating up/down monomer moments. (d–f) Schematic open chains with different terminations. $J_1$–$J_1$: two $S=1/2$ edge spins forming an edge triplet ($\sum_i \langle S_i^z \rangle \approx 1$). $J_2$–$J_2$: intact strong dimers and no free edges ($\sum_i \langle S_i^z \rangle \approx 0$). Mixed $J_1$–$J_2$: a single unpaired edge spin giving a Kramers doublet ($\sum_i \langle S_i^z \rangle \approx 1/2$).
• Figure 2: Topological diagnostics of the dimerized nanographene spin-$1/2$ chain. (a) Many-body Zak phase from twisted boundary conditions. $J_1$–$J_1$ (even $N$) gives $\pi$ (SPT), $J_2$–$J_2$ (even $N$) gives $0$ (trivial), and mixed $J_1$–$J_2$ (odd $N$) yields $\sim 0.75\pi$, consistent with a single fractional edge spin-$1/2$. (b) Entanglement spectra $-\log\lambda$ versus Schmidt index (with $\lambda$ the eigenvalues of the reduced density matrix) for cuts across weak ($J_1$) and strong ($J_2$) bonds. Weak-bond cuts in $J_1$–$J_1$ and $J_1$–$J_2$ chains show a nearly fourfold-degenerate lowest manifold (SPT), whereas a strong-bond cut in $J_2$–$J_2$ chains yields a non-degenerate spectrum (trivial). (c) Low-energy multiplet ladders from ED: $J_1$–$J_1$ has a triplet ground state (two correlated edge spins), $J_2$–$J_2$ a singlet (no free edges), and odd-$N$$J_1$–$J_2$ a Kramers doublet (single edge spin-$1/2$). (d) Spin-density maps $\langle S_i^z\rangle$ for the lowest multiplets, highlighting two edge-localized $S=1/2$ moments in $J_1$–$J_1$, no edge polarization in $J_2$–$J_2$, and a single edge $S=1/2$ in $J_1$–$J_2$.
• Figure 3: Spin-1 Hund chain from N-doped CTP superatoms and its Haldane signatures. (a) Chemical route from N-doped CTP to an $S=1$ Hund monomer and its one-dimensional polymer, with intra-monomer ferromagnetic $J_{\mathrm{FM}}$ and inter-monomer antiferromagnetic $J_{\mathrm{AFM}}$ couplings defining the spin-1 chain. (b) Simulated spin-resolved STM image of the isolated $S=1$ monomer, showing complementary $\alpha$- and $\beta$-spin contrast at the two radical sites (red arrows). (c) DFT+$U$ local magnetization for a polymer segment, revealing bulk antiferromagnetic order with alternating monomer moments. (d) Even-$N$ open chain with $J_{\mathrm{FM}}$–$J_{\mathrm{FM}}$ termination and the corresponding spin-$1/2$ lattice cartoon, exposing two edge $S=1/2$ moments that form an edge triplet. (e) Edge-triplet splitting $\Delta_{\mathrm{edge}}(L)$ versus chain length $L$ with an exponential fit $\Delta_{\mathrm{edge}}(L)=A\exp(-L/\xi)$, yielding a short correlation length of order two spin-1 monomers. (f) DMRG diagnostics: many-body Zak phase quantized to $\pi$, a nearly fourfold-degenerate entanglement/multiplet manifold, and spin-density maps $\langle S_i^z\rangle$ showing two localized $S=1/2$ edge spins with a slight intra-monomer asymmetry.
• Figure 4: (a,b) Spin-resolved DFT+$U$ band structures of the spin-$1/2$ and spin-1 chains in AFM (left) and FM (right) configurations (blue/red: opposite spins; dashed line: $E_F$, bands are spin-degenerate in AFM case). (c) One-orbital Hubbard cartoon: the spin-$1/2$ chain has one half-filled orbital per monomer with a single hopping $t$ and on-site interaction $U_{\mathrm{eff}} = 1.26$ eV, while in the spin-1 chain two ferromagnetically locked orbitals form an $S=1$ site with the same $t$ and $U_{\mathrm{eff}}$ and strong intra-monomer exchange $J_{\mathrm{FM}}$. (d) Bulk $S^{+-}(q,\omega)$ for the dimerized spin-$1/2$ chain, showing a very narrow W-shaped triplon band with bandwidth $\Delta W \approx 0.11$ meV $\approx J_1$; the dashed line is the one-triplon dispersion of the fitted spin model. (e) $S^{+-}(q,\omega)$ for the spin-1 Hund chain, displaying an M-shaped triplon branch with Haldane gap $\Delta E_H \approx 52$ meV $\approx 0.41 J_{\mathrm{AFM}}$, total bandwidth $\Delta W \approx 305$ meV $\approx 2.4 J_{\mathrm{AFM}}$, and additional weak branches from internal Hund modes; dashed lines again show the one-triplon dispersion.